396 BELL SYSTEM TECHNICAL JOURNAL 



namely, 



G(1000, L) = e" ^;,G(1000, L,). (10) 



If the type of complex tone can be chosen so that bk is unity and also 

 so that the values of Lk for each component are equal, then the funda- 

 mental equation for calculating loudness becomes 



GiL) = wG(L,), (14) 



where n is the number of components. Since we are always dealing 

 in this section with 6^(1000, L) or G(1000, Lk), the 1000 is left out in 

 the above nomenclature. If experimental measurements of L corre- 

 sponding to values of Lk are taken for a tone fulfilling the above con- 

 ditions throughout the audible range, the function G can be determined. 

 If we accept the theory that, when two simple tones widely separated 

 in frequency act upon the ear, the nerv^e terminals stimulated by each 

 are at different portions of the basilar membrane, then we would 

 expect the interference of the loudness of one upon that of the other 

 would be negligible. Consequently, for such a combination b is unity. 

 Measurements were made upon two such tones, the two components 

 being equally loud, the first having frequencies of 1000 and 2000 

 cycles and the second, frequencies of 125 and 1000 cycles. The ob- 

 served points are shown along the second curve from the top of Fig. 6. 

 The abscissae give the loudness level Lk of each component and the 

 ordinates the loudness level L of the two components combined. The 

 equation G(y) = 2G{x) should represent these data. Similar measure- 

 ments were made with a complex tone having 10 components, all 

 equally loud. The method of generating such tones is described in 

 Appendix C. The results are shown by the points along the top 

 curve of Fig. 6. The equation G(y) = lOG(x) should represent these 

 data except at high levels where bk is not unity. 



There is probably a complete separation between stimulated patches 

 of nerve endings when the first component is introduced into one ear 

 and the second component into the other ear. In this case the same 

 or different frequencies can be used. Since it is easier to make loud- 

 ness balances when the same kind of sound is used, measurements were 

 made (1) with 125-cycle tones (2) with 1000-cycle tones and (3) with 

 4000-cycle tones. The results are shown on Fig. 7. In this curve the 

 ordinates give the loudness levels when one ear is used while the 

 abscissae give the corresponding loudness levels for the same intensity 

 level of the tone when both ears are used for listening. If binaural 

 versus monaural loudness data actually fit into this scheme of calcula- 



